[9e3c33] | 1 | /* emacs edit mode for this file is -*- C++ -*- */ |
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[71da5e] | 2 | /* $Id: sm_util.cc,v 1.5 1998-03-10 14:48:34 schmidt Exp $ */ |
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[9e3c33] | 3 | |
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| 4 | //{{{ docu |
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| 5 | // |
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| 6 | // sm_util.cc - utlities for sparse modular gcd. |
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| 7 | // |
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| 8 | // Dependencies: Routines used by and only by sm_sparsemod.cc. |
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| 9 | // |
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[9ede7e] | 10 | // Contributed by Marion Bruder <bruder@math.uni-sb.de>. |
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| 11 | // |
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[9e3c33] | 12 | //}}} |
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| 13 | |
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| 14 | #include <config.h> |
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| 15 | |
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| 16 | #include "assert.h" |
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| 17 | #include "debug.h" |
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| 18 | |
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| 19 | #include "cf_defs.h" |
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[fbefc9] | 20 | #include "cf_algorithm.h" |
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[9e3c33] | 21 | #include "cf_iter.h" |
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| 22 | #include "cf_reval.h" |
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| 23 | #include "canonicalform.h" |
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| 24 | #include "variable.h" |
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[71da5e] | 25 | #include "ftmpl_array.h" |
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[9e3c33] | 26 | |
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| 27 | //{{{ static CanonicalForm fmonome( const CanonicalForm & f ) |
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| 28 | //{{{ docu |
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| 29 | // |
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| 30 | // fmonome() - return the leading monomial of a poly. |
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| 31 | // |
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| 32 | // As in Leitkoeffizient(), the leading monomial is calculated |
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| 33 | // with respect to inCoeffDomain(). The leading monomial is |
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| 34 | // returned with coefficient 1. |
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| 35 | // |
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| 36 | //}}} |
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| 37 | static CanonicalForm |
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| 38 | fmonome( const CanonicalForm & f ) |
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| 39 | { |
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| 40 | if ( f.inCoeffDomain() ) |
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| 41 | { |
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| 42 | return 1; |
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| 43 | } |
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| 44 | else |
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| 45 | { |
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| 46 | CFIterator J = f; |
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| 47 | CanonicalForm result; |
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| 48 | result = power( f.mvar() , J.exp() ) * fmonome( J.coeff() ); |
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| 49 | return result; |
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| 50 | } |
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| 51 | } |
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| 52 | //}}} |
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| 53 | |
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| 54 | //{{{ static CanonicalForm interpol( const CFArray & values, const CanonicalForm & point, const CFArray & points, const Variable & x, int d, int CHAR ) |
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| 55 | //{{{ docu |
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| 56 | // |
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[9ede7e] | 57 | // interpol() - Newton interpolation. |
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| 58 | // |
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| 59 | // Calculate f in x such that f(point) = values[1], |
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| 60 | // f(points[i]) = values[i], i=2, ..., d+1. |
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| 61 | // Make sure that you are calculating in a field. |
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| 62 | // |
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| 63 | // alpha: the values at the interpolation points (= values) |
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| 64 | // punkte: the point at which we interpolate (= (point, points)) |
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[9e3c33] | 65 | // |
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| 66 | //}}} |
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| 67 | static CanonicalForm |
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| 68 | interpol( const CFArray & values, const CanonicalForm & point, const CFArray & points, const Variable & x, int d, int CHAR ) |
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| 69 | { |
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| 70 | CFArray alpha( 1, d+1 ); |
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| 71 | int i; |
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| 72 | for ( i = 1 ; i <= d+1 ; i++ ) |
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| 73 | { |
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| 74 | alpha[i] = values[i]; |
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| 75 | } |
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| 76 | |
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| 77 | int k, j; |
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| 78 | CFArray punkte( 1 , d+1 ); |
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| 79 | for ( i = 1 ; i <= d+1 ; i++ ) |
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| 80 | { |
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| 81 | if ( i == 1 ) |
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| 82 | { |
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| 83 | punkte[i] = point; |
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| 84 | } |
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| 85 | else |
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| 86 | { |
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| 87 | punkte[i] = points[i-1]; |
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| 88 | } |
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| 89 | } |
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| 90 | |
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[9ede7e] | 91 | // calculate Newton coefficients alpha[i] |
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[9e3c33] | 92 | for ( k = 2 ; k <= d+1 ; k++ ) |
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| 93 | { |
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| 94 | for ( j = d+1 ; j >= k ; j-- ) |
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| 95 | { |
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| 96 | alpha[j] = (alpha[j] - alpha[j-1]) / (punkte[j] - punkte[j-k+1]); |
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| 97 | } |
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| 98 | } |
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| 99 | |
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[9ede7e] | 100 | // calculate f from Newton coefficients |
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[9e3c33] | 101 | CanonicalForm f = alpha [1], produkt = 1; |
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| 102 | for ( i = 1 ; i <= d ; i++ ) |
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| 103 | { |
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| 104 | produkt *= ( x - punkte[i] ); |
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[9ede7e] | 105 | f += ( alpha[i+1] * produkt ) ; |
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[9e3c33] | 106 | } |
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| 107 | |
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| 108 | return f; |
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| 109 | } |
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| 110 | //}}} |
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| 111 | |
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| 112 | //{{{ int countmonome( const CanonicalForm & f ) |
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| 113 | //{{{ docu |
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| 114 | // |
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| 115 | // countmonome() - count the number of monomials in a poly. |
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| 116 | // |
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| 117 | // As in Leitkoeffizient(), the number of monomials is calculated |
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| 118 | // with respect to inCoeffDomain(). |
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| 119 | // |
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| 120 | //}}} |
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| 121 | int |
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| 122 | countmonome( const CanonicalForm & f ) |
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| 123 | { |
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| 124 | if ( f.inCoeffDomain() ) |
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| 125 | { |
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| 126 | return 1; |
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| 127 | } |
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| 128 | else |
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| 129 | { |
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| 130 | CFIterator I = f; |
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| 131 | int result = 0; |
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| 132 | |
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| 133 | while ( I.hasTerms() ) |
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| 134 | { |
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| 135 | result += countmonome( I.coeff() ); |
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| 136 | I++; |
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| 137 | } |
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| 138 | return result; |
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| 139 | } |
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| 140 | } |
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| 141 | //}}} |
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| 142 | |
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| 143 | //{{{ CanonicalForm Leitkoeffizient( const CanonicalForm & f ) |
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| 144 | //{{{ docu |
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| 145 | // |
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| 146 | // Leitkoeffizient() - get the leading coefficient of a poly. |
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| 147 | // |
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| 148 | // In contrary to the method lc(), the leading coefficient is calculated |
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| 149 | // with respect to to the method inCoeffDomain(), so that a poly over an |
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| 150 | // algebraic extension will have a leading coefficient in this algebraic |
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| 151 | // extension (and *not* in its groundfield). |
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| 152 | // |
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| 153 | //}}} |
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| 154 | CanonicalForm |
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| 155 | Leitkoeffizient( const CanonicalForm & f ) |
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| 156 | { |
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| 157 | if ( f.inCoeffDomain() ) |
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| 158 | return f; |
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| 159 | else |
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| 160 | { |
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| 161 | CFIterator J = f; |
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| 162 | CanonicalForm result; |
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| 163 | result = Leitkoeffizient( J.coeff() ); |
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| 164 | return result; |
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| 165 | } |
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| 166 | } |
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| 167 | //}}} |
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| 168 | |
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| 169 | //{{{ void ChinesePoly( int arraylength, const CFArray & Polys, const CFArray & primes, CanonicalForm & result ) |
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| 170 | //{{{ docu |
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| 171 | // |
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| 172 | // ChinesePoly - chinese remaindering mod p. |
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| 173 | // |
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| 174 | // Given n=arraylength polynomials Polys[1] (mod primes[1]), ..., |
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| 175 | // Polys[n] (mod primes[n]), we calculate result such that |
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| 176 | // result = Polys[i] (mod primes[i]) for all i. |
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| 177 | // |
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| 178 | // Note: We assume that all monomials which occure in Polys[2], |
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| 179 | // ..., Polys[n] also occure in Polys[1]. |
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| 180 | // |
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| 181 | // bound: number of monomials of Polys[1] |
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| 182 | // mono: array of monomials of Polys[1]. For each monomial, we |
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| 183 | // get the coefficients of this monomial in all Polys, store them |
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| 184 | // in koeffi and do chinese remaindering over these coeffcients. |
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| 185 | // The resulting polynomial is constructed monomial-wise from |
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| 186 | // the results. |
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| 187 | // polis: used to trace through monomials of Polys |
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| 188 | // h: result of remaindering of koeffi[1], ..., koeffi[n] |
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| 189 | // Primes: do we need that? |
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| 190 | // |
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| 191 | //}}} |
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| 192 | void |
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| 193 | ChinesePoly( int arraylength, const CFArray & Polys, const CFArray & primes, CanonicalForm & result ) |
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| 194 | { |
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| 195 | DEBINCLEVEL( cerr, "ChinesePoly" ); |
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| 196 | |
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| 197 | CFArray koeffi( 1, arraylength ), polis( 1, arraylength ); |
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| 198 | CFArray Primes( 1, arraylength ); |
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| 199 | int bound = countmonome( Polys[1] ); |
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| 200 | CFArray mono( 1, bound ); |
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| 201 | int i, j; |
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| 202 | CanonicalForm h, unnecessaryforme; |
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| 203 | |
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| 204 | DEBOUTLN( cerr, "Interpolating " << Polys ); |
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| 205 | DEBOUTLN( cerr, "modulo" << primes ); |
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| 206 | for ( i = 1 ; i <= arraylength ; i++ ) |
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| 207 | { |
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| 208 | polis[i] = Polys[i]; |
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| 209 | Primes[i] = primes[i]; |
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| 210 | } |
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| 211 | |
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| 212 | for ( i = 1 ; i <= bound ; i++ ) |
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| 213 | { |
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| 214 | mono[i] = fmonome( polis[1] ); |
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| 215 | koeffi[1] = lc( polis[1] ); // maybe better use Leitkoeffizient ?? |
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| 216 | polis[1] -= mono[i] * koeffi[1]; |
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| 217 | for ( j = 2 ; j <= arraylength ; j++ ) |
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| 218 | { |
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| 219 | koeffi[j] = lc( polis[j] ); // see above |
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| 220 | polis[j] -= mono[i] * koeffi[j]; |
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| 221 | } |
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| 222 | |
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| 223 | // calculate interpolating poly for each monomial |
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| 224 | chineseRemainder( koeffi, Primes, h, unnecessaryforme ); |
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| 225 | result += h * mono[i]; |
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| 226 | } |
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| 227 | DEBOUTLN( cerr, "result = " << result ); |
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| 228 | |
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| 229 | DEBDECLEVEL( cerr, "ChinesePoly" ); |
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| 230 | } |
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| 231 | //}}} |
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| 232 | |
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| 233 | //{{{ CanonicalForm dinterpol( int d, const CanonicalForm & gi, const CFArray & zwischen, const REvaluation & alpha, int s, const CFArray & beta, int ni, int CHAR ) |
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| 234 | //{{{ docu |
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| 235 | // |
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| 236 | // dinterpol() - calculate interpolating poly (???). |
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| 237 | // |
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| 238 | // Calculate f such that f is congruent to gi mod (x_s - alpha_s) and |
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| 239 | // congruent to zwischen[i] mod (x_s - beta[i]) for all i. |
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| 240 | // |
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| 241 | //}}} |
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| 242 | CanonicalForm |
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| 243 | dinterpol( int d, const CanonicalForm & gi, const CFArray & zwischen, const REvaluation & alpha, int s, const CFArray & beta, int ni, int CHAR ) |
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| 244 | { |
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| 245 | int i, j, lev = s; |
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| 246 | Variable x( lev ); |
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| 247 | |
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| 248 | CFArray polis( 1, d+1 ); |
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| 249 | polis[1] = gi; |
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| 250 | for ( i = 2 ; i <= d+1 ; i++ ) |
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| 251 | { |
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| 252 | polis[i] = zwischen[i-1]; |
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| 253 | } |
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| 254 | |
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| 255 | CFArray mono( 1, ni ), koeffi( 1, d+1 ); |
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| 256 | CanonicalForm h , f = 0; |
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| 257 | |
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| 258 | for ( i = 1 ; i <= ni ; i++ ) |
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| 259 | { |
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| 260 | mono[i] = fmonome( polis[1] ); |
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| 261 | |
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| 262 | koeffi[1] = Leitkoeffizient( polis[1] ); |
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| 263 | polis[1] -= mono[i] * koeffi[1]; |
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| 264 | |
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| 265 | for ( j = 2 ; j <= d+1 ; j++ ) |
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| 266 | { |
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| 267 | koeffi[j] = Leitkoeffizient( polis[j] ); |
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| 268 | polis[j] -= mono[i] * koeffi[j]; |
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| 269 | } |
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| 270 | |
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| 271 | // calculate interpolating poly for each monomial |
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| 272 | h = interpol( koeffi, alpha[s] , beta, x , d , CHAR ); |
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| 273 | |
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| 274 | f += h * mono[i]; |
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| 275 | } |
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| 276 | |
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| 277 | return f; |
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| 278 | } |
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| 279 | //}}} |
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| 280 | |
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| 281 | //{{{ CanonicalForm sinterpol( const CanonicalForm & gi, const Array<REvaluation> & xi, CanonicalForm* zwischen, int n ) |
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| 282 | //{{{ docu |
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| 283 | // |
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| 284 | // sinterpol - sparse interpolation (???). |
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| 285 | // |
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| 286 | // Loese Gleichungssystem: |
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| 287 | // x[1], .., x[q]( Tupel ) eingesetzt fuer die Variablen in gi ergibt |
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| 288 | // zwischen[1], .., zwischen[q] |
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| 289 | // |
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| 290 | //}}} |
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| 291 | CanonicalForm |
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| 292 | sinterpol( const CanonicalForm & gi, const Array<REvaluation> & xi, CanonicalForm* zwischen, int n ) |
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| 293 | { |
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| 294 | CanonicalForm f = gi; |
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| 295 | int i, j; |
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| 296 | CFArray mono( 1 , n ); |
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| 297 | |
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| 298 | // mono[i] is the i'th monomial |
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| 299 | for ( i = 1 ; i <= n ; i++ ) |
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| 300 | { |
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| 301 | mono[i] = fmonome( f ); |
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| 302 | f -= mono[i]*Leitkoeffizient(f); |
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| 303 | } |
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| 304 | |
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| 305 | // fill up matrix a |
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| 306 | CFMatrix a( n , n + 1 ); |
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| 307 | for ( i = 1 ; i <= n ; i++ ) |
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| 308 | for ( j = 1 ; j <= n + 1 ; j++ ) |
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| 309 | { |
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| 310 | if ( j == n+1 ) |
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| 311 | { |
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| 312 | a[i][j] = zwischen[i]; |
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| 313 | } |
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| 314 | else |
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| 315 | { |
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| 316 | a[i][j] = xi[i]( mono[j] ); |
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| 317 | } |
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| 318 | } |
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| 319 | |
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| 320 | // sove a*y=zwischen and get soultions y1, .., yn mod p |
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| 321 | linearSystemSolve( a ); |
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| 322 | |
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| 323 | for ( i = 1 ; i <= n ; i++ ) |
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| 324 | f += a[i][n+1] * mono[i]; |
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| 325 | |
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| 326 | return f; |
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| 327 | } |
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| 328 | //}}} |
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